m.i. jaime alfonso reyes ´cortés. the basic task for any probabilistic inference system is to...
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BAYESIAN NETWORKS (CONT.)M.I. Jaime Alfonso Reyes ´Cortés
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INFERENCE IN BAYESIAN NETWORKS
The basic task for any probabilistic inference system is to compute the posterior probability distribution for a set of query nodes, given values for some evidence nodes.
This task is called belief updating or probabilistic inference.
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EXACT INFERENCE IN CHAINS
Two node network A two node network If there is evidence about the parent node,
say X = x, then the posterior probability (or belief) for Y can be read straight from the value in CPT
If there is evidence about the child node, say Y = y, then the inference task of updating the belief for X is done using
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P(x) is the prior and l(x) = P(Y = y| X= x) is the likelihood
Note that we don’t need to know the prior for the evidence. Since the beliefs for all the values of X must sum to one (due to the Total Probability) , we can compute a as a normalizing constant
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Three node chain If we have evidence about the root
node, X=x, updating in the same direction as the arcs involves the simple application of the chain rule, using the independencies implied in the network
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If we have evidence about the leaf node, Z=z, the diagnostic inference to obtain Bel(X) is done with the application of Bayes’ Theorem and the chain rule
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Exact inference in polytrees polytree (or “forest”) Polytrees have at most one path between any
pair of nodes; hence they are also referred to as singly-connected networks
Assume X is the query node, and there is some set of evidence nodes E (not including X)
The task is to update Bel(X) by computing P(X|E) The local belief updating for X must incorporate
evidence from all other parts of the network
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evidence can be divided into:
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EJEMPLO
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REFERENCIAS
Kevin B. Korb, Ann E. Nicholson. Bayesian Artificial Intelligence, CHAPMAN & HALL/CRC. England, 2004.